Maths Olympiad Prep

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Combinatorics Difficulty 4.9 AIME Find the answer

4. There are student Ling † Ren who need accommodation. If each room houses 4 people, then there are still 20 people without accommodation; if each room houses 8 people, then there is one room that is neither empty nor full. Therefore, the number of dormitory rooms is \qquad rooms, and the number of students is \qquad people.

A number or a short expression. Spacing and $ signs are ignored.

Solution

4.6,444.6,44.
Let the number of dormitories be xx, and the number of students be yy (both x,yx, y are integers). Then
\left\{\begin{array} { l } { y - 2 0 5 . \end{array}\right.\right.

Therefore, x=6,y=44x=6, y=44.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.